I'd like to solve a generalized eigenvalue problem of the form:

$$\mathrm{A}x = \lambda \mathrm{B}x$$ $$s.t. x_i^T\mathrm{B}x_i=1.$$

Where $\mathrm{A}$ and $\mathrm{B}$ are symmetric but low-rank matrices and $x_i$ is the eigenvector associated with the largest eigenvalue $\lambda_i$.

I've been playing around with the geigen package in R, which provides a nice wrapper to LAPACK routines, and it seems that the constraint is satisfied only when either $\mathrm{A}$ or $\mathrm{B}$ is positive definite (I'm assuming it is implementing a Cholesky decomposition).

In other cases, geigen default's to LAPACK's dggev routine which results in eigenvectors that are orthogonal to $\mathrm{B}$. That is, $x_i\mathrm{B}x_i=0$, and so scaling/normalizing the eigenvectors have no effect.

I don't know enough to determine if this is expected behavior or if there another implementation to this generalized eigenvalue problem which satisfies this constraint?



1 Answer 1


Answering my own question here thanks to this paper and some careful Googling, I get the desired result thanks to simultaneous diagonalization, with two applications of SVD. I've adapted the notation to be consistent with my original question.

$$\mathrm{X}^T\mathrm{A}\mathrm{X}=\Lambda$$ $$\mathrm{X}^T\mathrm{B}\mathrm{X}=\mathrm{I}$$

The goal is to find the transformation matrix $\mathrm{X}$ and diagonal matrix $\Lambda$, so that the columns contain the eigenvectors and diagonal entries contain eigenvalues of the generalized eigenvalue problem, respectively.

First perform a singular value decomposition of $\mathrm{B}$:

$$\mathrm{B}=\mathrm{V}_r\Sigma_r\mathrm{V}_r^T\equiv \mathrm{X}'(\mathrm{X}')^T$$

where $\mathrm{X}' \equiv \mathrm{V}_r\Sigma_r^{-\frac{1}{2}}$ and $r=\mathrm{rank}(\mathrm{B})$, so we're taking the first $r$ singular values and vectors to define matrix $\mathrm{X}'$.

Next, define a matrix $\mathrm{C}=(\mathrm{X}')^T\mathrm{A}\mathrm{X}'$ and perform an svd to obtain:

$$\mathrm{C}=(\mathrm{X}'')^T\Sigma' \mathrm{X}''$$

then $\mathrm{X} = \mathrm{X}'\mathrm{X}''$ and the above equations are satisfied if $\mathrm{I}$ and $\Lambda$ are of dimension $r\times r$.

  • $\begingroup$ not sure if this implementation is the most efficient though, since it involves several matrix multiplications and 2 rounds of SVD. $\endgroup$
    – zzk
    Jan 16, 2015 at 16:54

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